If I'm doing this right, the Anarchy Zones is a pretty big world with a lot of options, quite possibly too big with difficulty finding where to begin.
Think small. Think Dread.
Yes, there are some pretty big things about to happen in the zone. A least one city-state is planning for war, another desperate to find resources, and a third rife with internal conflict about its cybernetic dreams. No one knows if or when another alien ship will show up. But all these are symptoms, cogs of the grand plan, and not the machine itself.
Being elite members of a city-state militia sent on a mission to another place is certainly one way to bring a group together and give them a goal. Its unlikely for a starting character though.
Just getting food, or some money for the family, or finding supplies for this year's harvest are all good points. It is a personal stake, rather than being a go-fer in the plans of someone else, and a lot closer to heart. Now the struggle is do they sell out to some other cause to achieve this. Joining the militia gets some money, but sends the character into danger. Sneaking into cordoned off areas of the arcology can get supplies, but is highly illegal.
In short, don't try to change the zone - at least not at first. Instead, ask how the zone will change the characters, and keep a focus on that.
The Anarchy Zones isn't the usual - "something is going to try to eat my face in the next few moments" horror. Any sieges of a mall between a few thousand reanimates and a handful of survivors ended years ago, and poorly for the survivors. But there is still a chance of such event happening again if you aren't fast enough, or smart enough. A shortage of bullets is a lot less important than a shortage of food.
Although not alive, the Zone is rather akin to the otherworldly horrors of HP Lovecraft. Conflict is often less will a person live to the next day, than why. People in the city states will back-stab you for more resources, the government doesn't seem to be acting, raiders are roaming, you can never tell what the alien's motives are - what is a person to do?
Planetary Citizens are smart and well equipped opponents, if not always looking for a fight. Actual humans are probably the most ruthless, and most aware of what is to gain. Reanimates are smarter than a simple Romero style zombie. Characters are just a small part of the world, yet they may have to face all three of these things to fulfill even simple desires.